Review of "Spectral and Dynamical Stability of Nonlinear Waves" by T Kapitula and K Promislow

By Bjorn Sandstede
Print
Spectral and Dynamical Stability of Nonlinear Waves
by Todd Kapitula and Keith Promislow
Springer-Verlag
Applied Mathematical Sciences Series, Vol 185
361pp. (2013)
ISBN 978-1-4614-6995-7
Reviewed by Bjorn Sandstede
Applied Mathematics
Brown University

This 368-page book provides an excellent introduction to the spectral and nonlinear stability theory of nonlinear waves in one-dimensional domains. It is aimed primarily at graduate students, but it can certainly also be used by postdocs and other researchers who are interested in learning more about this area.

The book is quite comprehensive and broad in scope: both spectral and nonlinear stability results are covered, and the authors discuss both dissipative and Hamiltonian systems. More specifically, details on the essential and point spectra of pulses, fronts, and wave trains are provided. Two chapters focus on nonlinear stability theorems: for dissipative systems, the authors use a $C^0$ semigroup framework on Hilbert spaces; for Hamiltonian systems, the constrained-minimization approach by Grillakis, Shatah, and Strauss is covered. Finally, various techniques for calculating the point spectrum of nonlinear waves are outlined: the authors cover perturbation and reduction methods as well as Krein signatures of eigenvalues and give an extensive introduction to the Evans function, both on bounded intervals and the unbounded real line, including its analytic extension across the essential spectrum.

There are several features that make this book accessible and ready for use in a graduate-level topics course.

First, the authors included a well-written and comprehensive chapter that provides background material on linear ordinary differential equations, functional analysis (closed operator, Fredholm theory), and Sturm-Liouville theory. Some of the results are proved, and detailed references are given for all statements that are quoted without a proof.

Second, the authors focus on nonlinear waves for nth-order partial differential equations (as opposed to systems of PDEs). This simplifies much of the exposition without losing too much generality (the extensions necessary for systems are mentioned, but not worked out in detail).

Third, the book contains many examples in which the theoretical results are applied to concrete PDEs: including these examples, and returning to them throughout the book to explain and illustrate the various different results and approaches covered, makes the book very readable and more applicable.

Finally, each section ends with a number of exercises of different levels and different scope. Some of these involve proving some of the statements; others are concerned will applying techniques to specific PDEs.

While the book is quite comprehensive, there are a few topics and techniques that are not covered. For instance, extensions to multi-dimensional domains are mentioned only in passing, and the connections between spectral results and the dynamical properties of the underlying traveling-wave equations are not always elucidated. Not much guidance is given to readers who would like to complement the theoretical analyses with numerical computations (although the authors do include a discussion of absolute spectra and their relevance for spectra of differential operators on large but bounded domains). But these are minor points that more likely illustrate the different preferences of the authors and those of this reviewer.

Overall this book covers a broad range of topics in an area for which not many alternatives exist: the book is, in my opinion, an excellent addition to the literature.

Categories: Magazine, Book Reviews
Tags:

Please login or register to post comments.

Name:
Email:
Subject:
Message:
x